BPS - 3rd Ed. Chapter 6 1

Chapter 6

Two-Way Tables

BPS - 3rd Ed. Chapter 6 2

In prior chapters we studied the relationship between two quantitative variables with – Correlation – Regression

In this chapter we study the relationship between two categorical variables using– Counts– Marginal percents– Conditional percents

Categorical Variables

BPS - 3rd Ed. Chapter 6 3

Data are cross-tabulated to form a two-way table with a row variable and column variable

The count of observations falling into each combination of categories is cross-tabulated into each table cell

Counts are totaled to create marginal totals

Two-Way Tables

BPS - 3rd Ed. Chapter 6 4

Case Study

Data from the U.S. Census Bureau (2000)

Level of education by age

(Statistical Abstract of the United States, 2001)

Age and Education

BPS - 3rd Ed. Chapter 6 5

Case StudyAge and Education

Variables

Marginal distributions

BPS - 3rd Ed. Chapter 6 6

Case StudyAge and Education

Variables

Marginal totals

37,786 81,435 56,008

27,85858,07744,46544,828

BPS - 3rd Ed. Chapter 6 7

It is more informative to display counts as percents

Marginal percents

Use a bar graph to display marginal percents (optional)

Marginal Percents

%100 totaltable

totalmarginal percent marginal

100% totaltable

totalmarginal percent marginal

BPS - 3rd Ed. Chapter 6 8

Case StudyAge and Education

Row Marginal Distribution

Did not graduate HS

27,859 ÷ 175,230 × 100% = 15.9%

Did graduate HS

58,077 / 175,230 × 100% = 33.1%

Finished 1-3 yrs college

44,465 / 175,230 × 100% = 25.4%

Finished ≥4 yrs college

44,828 / 175,230 × 100% = 25.6%

BPS - 3rd Ed. Chapter 6 9

Relationships are described with conditional percents

There are two types of conditional percents:– Column percents– Row percents

Conditional Percents

BPS - 3rd Ed. Chapter 6 10

Row Conditional Percent Column Conditional Percent

100%alcolumn tot

count cell cellfor percent column

100% totalrowcount cell cellfor percent row

To know which to use, ask “What comparison is most relevant?”

BPS - 3rd Ed. Chapter 6 11

Case StudyAge and Education

Compare the 25-34 age group to the 35-54 age group in % completing college:

Change the counts to column percents (important):

group age 54-35 for (28.4%) .28481,43523,160

group age 34-25 for (29.3%) .29337,78611,071

BPS - 3rd Ed. Chapter 6 12

Case StudyAge and Education

If we compute the percent completing college for all of the age groups, this gives conditional distribution (column percents) completing college by age:

Age: 25-34 35-54 55 and over

Percent with≥ 4 yrs college: 29.3% 28.4% 18.9%

BPS - 3rd Ed. Chapter 6 13

If the conditional distributions are nearly the same, then we say that there is not an association between the row and column variables

If there are significant differences in the conditional distributions, then we say that there is an association between the row and column variables

Association

BPS - 3rd Ed. Chapter 6 14

Column Percents for College DataFigure 6.2 (in text)

Negative association -- higher age had lower rate of Coll. Graduation

BPS - 3rd Ed. Chapter 6 15

Simpson’s paradox a lurking variable creates a reversal in the direction of the association

To uncover Simpson’s Paradox, divide data into subgroups based on the lurking variable

Simpson’s Paradox

BPS - 3rd Ed. Chapter 6 16

Consider college acceptance rates by sex

Discrimination? (Simpson’s Paradox)

Accepted Notaccepted Total

Men 198 162 360

Women 88 112 200

Total 286 274 560

198 of 360 (55%) of men accepted 88 of 200 (44%) of women accepted Is this discrimination?

BPS - 3rd Ed. Chapter 6 17

Discrimination? (Simpson’s Paradox)

Or is there a lurking variable that explains the association?

To evaluate this, split applications according to the lurking variable “School applied to”– Business School (240 applicants) – Art School (320 applicants)

BPS - 3rd Ed. Chapter 6 18

Discrimination? (Simpson’s Paradox)

18 of 120 men (15%) of men were accepted to B-school24 of 120 (20%) of women were accepted to B-schoolA higher percentage of women were accepted

BUSINESS SCHOOL

Accepted Notaccepted Total

Men 18 102 120

Women 24 96 120

Total 42 198 240

BPS - 3rd Ed. Chapter 6 19

Discrimination (Simpson’s Paradox)

ART SCHOOL

180 of 240 men (75%) of men were accepted64 of 80 (80%) of women were accepted A higher percentage of women were accepted.

Accepted Notaccepted Total

Men 180 60 240

Women 64 16 80

Total 244 76 320

BPS - 3rd Ed. Chapter 6 20

Within each school, a higher percentage of women were accepted than men. (There was not any discrimination against women.)

This is an example of Simpson’s Paradox. – When the lurking variable (School applied to) was

ignored, the data suggest discrimination against women.

– When the School applied to was considered, the association is reversed.

Discrimination? (Simpson’s Paradox)